Translation functors for locally analytic representations

TL;DR

This paper develops translation functors for locally analytic representations of p-adic Lie groups, establishing their properties, equivalences, and connections to category O and p-adic Langlands correspondence.

Contribution

It introduces and analyzes translation functors for coadmissible modules over the locally analytic distribution algebra, linking them to category O and p-adic Langlands representations.

Findings

Translation functors induce equivalences between subcategories.

Connections established between translation functors and category O.

Effects on locally analytic representations related to p-adic Langlands correspondence.

Abstract

Let $G$ be a $p$-adic Lie group with reductive Lie algebra $\mathfrak{g}$. In analogy to the translation functors introduced by Bernstein and Gelfand on categories of $U(\mathfrak{g})$-modules we consider similarly defined functors on the category of coadmissible modules over the locally analytic distribution algebra $D(G)$ on which the center of $U(\mathfrak{g})$ acts locally finite. These functors induce equivalences between certain subcategories of the latter category. Furthermore, these translation functors are naturally related to those on category $\mathcal{O}$ via the functors from category $\mathcal{O}$ to the category of coadmissible modules. We also investigate the effect of the translation functors on locally analytic representations $\Pi(V)^{\rm la}$ associated by the $p$-adic Langlands correspondence for ${\rm GL}_2(\mathbb{Q}_p)$ to 2-dimensional Galois representations $V$.